Résume | The lifting method is a promising program to classify finite-dimensional pointed Hopf algebras. In fact, Andruskiewitsch and Schneider [AS] have used it to obtain the family of all pointed Hopf algebras over the complex numbers, whose coradical is the group algebra of an abelian group of order coprime with 210. It uses the classification of braidings of diagonal type obtained by Heckenberger [H].
In this talk we will present the advances in the general framework, for abelian groups of arbitrary order. It involves the consideration of the Weyl groupoid and a generalized root system for Nichols algebras of diagonal type, coideal subalgebras classified by Heckenberger and Schneider, and convex orders on these roots [A1], in order to have a family of relations between the generators of a PBW basis. A second step involves a minimal presentation by generators and relations needed to prove that any finite dimensional pointed Hopf algebra is generated by group-like elements and skew-primitive elements [A2]. The last step is the computation of all the liftings (that is, all the Hopf algebras associated to a fixed coradically graded Hopf algebras) and uses also this minimal presentation; it is part of a work in progress with N. Andruskiewitsch and A. García Iglesias.
[AS] N. Andruskiewitsch and H.-J. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. Math. 171(1) (2010), 375--417.
[A1] I. Angiono, A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems, JEMS, to appear.
[A2] I. Angiono, On Nichols algebras of diagonal type. J. Reine Angew. Math., to appear.
[H] I. Heckenberger, Classification of arithmetic root systems, Adv. Math. 220 (2009), 59--124. |